Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T01:14:43.918Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Subfields of Qp, and their Non-Standard Analogues

Published online by Cambridge University Press:  20 November 2018

Diana L. Dubrovsky
Diana L. Dubrovsky*
Affiliation:
Université de Montréal, Montréal, Québec
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The desire to study constructive properties of given mathematical structures goes back many years; we can perhaps mention L. Kronecker and B. L. van der Waerden, two pioneers in this field. With the development of recursion theory it was possible to make precise the notion of "effectively carrying out" the operations in a given algebraic structure. Thus, A. Frölich and J. C. Shepherdson [7] and M. O . Rabin [13] studied computable algebraic structures, i.e. structures whose operations can be viewed as recursive number theoretic relations. A. Robinson [18] and E. W. Madison [11] used the concepts of computable and arithmetically definable structures in order to establish the existence of what can be called non-standard analogues (in a sense that will be specified later) of certain subfields of R and C, the standard models for the theories of real closed and algebraically closed fields respectively.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 2 , April 1974 , pp. 473 - 491
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Ax, J. and Kochen, S., Diophantine problems over local fields, I, Amer. J. Math. 87 (1965), 605–630.Google Scholar
2. Ax, J. and Kochen, S., Diophantine problems over local fields, II, Amer. J. Math. 87 (1965), 631–648.Google Scholar
3. Ax, J. and Kochen, S., Diophantine problems over local fields, III, Ann. of Math. 83 (1966), 437–456.Google Scholar
4. Cohen, P. J., Decision procedures for real and p-adic fields, Comm. Pure Appl. Math. 22 (1969), 131–151.Google Scholar
5. Ershov, Yu. L., Numbered fields, Logic, Methodology and the Philosophy of Science, Proceedings of the 1967 International Congress (North-Holland Publishing Co., Amsterdam, 1968), 31–34.Google Scholar
6. Ershov, Yu. L., On the elementary theory of maximal normed fields (Russian), Dokl. Akad. Nauk. SSSR 165 (1965), 21–23; translated in Soviet Math. Dokl. 6 (1965), no. 6.Google Scholar
7. Frolich, A. and Shepherdson, J. C., Effective procedures in field theory, Trans. Roy. Soc. London Ser. A248 (1956), 407–432.Google Scholar
8. Kleene, S. C., Introduction to metamathematics (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1952).Google Scholar
9. Kochen, S., Integer-valued rational functions over the p-adic numbers: A p-adic analogue of the theory of real fields, “ Number Theory”, Amer. Math. Soc, Proc. of Symp. in Pure Math. 12 (1969), 57–73.Google Scholar
10. Lachlan, A. H. and Madison, E. W., Computable fields and arithmetically definable ordered fields, Proc. Amer. Math. Soc. 24 (1970), 803–807.Google Scholar
11. Madison, E. W., Computable algebraic structures and non-standard arithmetic, Trans. Amer. Math. Soc. 130 (1968), 38–54.Google Scholar
12. Nerode, A., A decision method for p-adic integral zeros of diophantine equations, Bull. Amer. Math. Soc. 69 (1963), 513–517.Google Scholar
13. Rabin, M. O., Computable algebra, Trans. Amer. Math. Soc. 95 (1960), 341–360.Google Scholar
14. Ribenboim, P., Théorie des valuations (University of Montreal Press, Montreal, 1964).Google Scholar
15. Rice, H. G., Recursive real numbers, Proc. Amer. Math. Soc. 5 (1954), 784–791.Google Scholar
16. Robinson, A., Complete theories (North-Holland Publishing Company, Amsterdam, 1956).Google Scholar
17. Robinson, A., Introduction to model theory and to the meta-mathematics of algebra (North-Holland Publishing Company, Amsterdam, 1963).Google Scholar
18. Robinson, A., Model theory and non-standard arithmetic, Infinitistic Methods (Symposium on Foundations of Mathematics), Warsaw, 1959.Google Scholar
19. Rogers, H., Theory of recursive functions and effective computability (McGraw-Hill Book Company, New York, 1967).Google Scholar
20. van der Waerden, B. L., Modern algebra, revised English edition (Frederick Unger Publishing Co., New York, 1953).Google Scholar